Question

The quantity $y$ is believed theoretically to depend linearly on the quantity $x$; that is $y=Ax+B$. Experimental results are $$\begin{array}{cccc}x& 1& 2& 3\\ y& 5\pm 2& 9\pm 1& 15\pm 2\end{array}$$ (a) Evaluate $A$ and $B$, with probable errors for each. (b) Evaluate $y(4)$ and its probable error.

Short answer

A = 4, B = 1 with probable errors ±1 each. y(4) = 17 with a probable error of ±4.12

Step-by-step solution

- Formulate the system of equations

Given the linear relationship, we can form equations using the given data points. We start with the equation: $$y=Ax+B$$. For each pair of $(x,y)$:$$\begin{array}{rlrl}{y}_1& =A{x}_1+B{y}_2& =A{x}_2+B{y}_3& =A{x}_3+B\end{array}$$. Plug in the values:$$\begin{array}{rlrl}5& =A\times 1+B(1)9& =A\times 2+B(2)15& =A\times 3+B(3)\end{array}$$

- Solve for A and B

Subtract equation (1) from equation (2) and (2) from (3) to eliminate $B$ and solve for $A$:$$\begin{array}{rlrl}(9-5)& =A(2-1)4& =A(1)A& =4\end{array}$$. Now use the value of $A$ in any of the initial equations to find $B$. Let's use equation (1):$$\begin{array}{rlrl}5& =4\times 1+B5& =4+BB& =1\end{array}$$

- Calculate probable errors for A and B

Average uncertainty = ($2\times 2+1\times 1+2\times 2$ ) / (5) = 1. Thus, both parameters carry this error.

- Calculate y(4)

Use the derived equation $y=4x+1$:$$\begin{array}{rlrl}y(4)& =4\times 4+1y(4)& =16+1y(4)& =17\end{array}$$. The probable error in $y(4)$ will propagate according to the linear sum of errors: Error(y(4)) = sqrt( (4 \times Error_A)^2 + (Error_B)^2 ) \ \ approx = sqrt( (4\times 1)^2 + 1^2) = sqrt(17) = 4.123

Key concepts

linear relationship

A linear relationship describes a direct proportionality between two variables. In this scenario, if we plot quantity y against quantity x, they will form a straight line, characterized by the equation: y = Ax + BHere, A represents the slope, indicating how much y changes for each unit increase in x. B is the y-intercept, showing the value of y when x is zero. This concept allows us to predict y for any given x if we know A and B.

experimental data analysis

Experimental data analysis involves examining data collected through experiments to deduce meaningful patterns or relationships. For this problem:

• We use the provided data points (x, y) to determine the constants A and B in the equation y = Ax + B.
• By substituting x and y pairs into the linear equation, we set up systems of equations to solve for A and B.

Next, using algebra, we isolate and solve these constants. Subtracting the equations properly helps in eliminating variable B to find A. Once A is known, it’s substituted back into one of the original equations to find B.

error propagation

Error propagation refers to how uncertainty in measurements affects the uncertainty in derived results. When performing calculations with experimental data, errors in the original measurements propagate according to specific rules. Here’s a quick snapshot:

• Average uncertainty combines individual errors from the data points.
• When we calculate y(4), its error needs to consider the errors of constants A and B.

Using the given formula, we can calculate the probable error in y(4):y(4) = 4x + 1If y(4) = 17 with calculated propagated error as sqrt( (4 * Error_A)^2 + (Error_B)^2 ) this becomes: sqrt( (4 * 1)^2 + 1^2 ) = sqrt(17) ≈ 4.123This lets us understand the measure of reliability in our calculated results.